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Theorem canthwdom 9613
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 9158, equivalent to canth 7367). (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
canthwdom ¬ 𝒫 𝐴* 𝐴

Proof of Theorem canthwdom
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elpw 5351 . . . . 5 ∅ ∈ 𝒫 𝐴
2 ne0i 4335 . . . . 5 (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅)
31, 2mp1i 13 . . . 4 (𝒫 𝐴* 𝐴 → 𝒫 𝐴 ≠ ∅)
4 brwdomn0 9603 . . . 4 (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴))
53, 4syl 17 . . 3 (𝒫 𝐴* 𝐴 → (𝒫 𝐴* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴))
65ibi 266 . 2 (𝒫 𝐴* 𝐴 → ∃𝑓 𝑓:𝐴onto→𝒫 𝐴)
7 relwdom 9600 . . . . 5 Rel ≼*
87brrelex2i 5730 . . . 4 (𝒫 𝐴* 𝐴𝐴 ∈ V)
9 foeq2 6802 . . . . . . 7 (𝑥 = 𝐴 → (𝑓:𝑥onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝑥))
10 pweq 4612 . . . . . . . 8 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
11 foeq3 6803 . . . . . . . 8 (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
1210, 11syl 17 . . . . . . 7 (𝑥 = 𝐴 → (𝑓:𝐴onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
139, 12bitrd 278 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑥onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
1413notbid 317 . . . . 5 (𝑥 = 𝐴 → (¬ 𝑓:𝑥onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴onto→𝒫 𝐴))
15 vex 3467 . . . . . 6 𝑥 ∈ V
1615canth 7367 . . . . 5 ¬ 𝑓:𝑥onto→𝒫 𝑥
1714, 16vtoclg 3534 . . . 4 (𝐴 ∈ V → ¬ 𝑓:𝐴onto→𝒫 𝐴)
188, 17syl 17 . . 3 (𝒫 𝐴* 𝐴 → ¬ 𝑓:𝐴onto→𝒫 𝐴)
1918nexdv 1932 . 2 (𝒫 𝐴* 𝐴 → ¬ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴)
206, 19pm2.65i 193 1 ¬ 𝒫 𝐴* 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1534  wex 1774  wcel 2099  wne 2930  Vcvv 3463  c0 4323  𝒫 cpw 4598   class class class wbr 5144  ontowfo 6542  * cwdom 9598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-wdom 9599
This theorem is referenced by:  pwdjudom  10248
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